A maximal Function Approach to Two-Measure Poincaré Inequalities

Juha Kinnunen, Riikka Korte, Juha Lehrbäck, Antti V. Vähäkangas*

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

This paper extends the self-improvement result of Keith and Zhong in Keith and Zhong (Ann. Math. 167(2):575–599, 2008) to the two-measure case. Our main result shows that a two-measure (p, p)-Poincaré inequality for 1 < p< ∞ improves to a (p, p- ε) -Poincaré inequality for some ε> 0 under a balance condition on the measures. The corresponding result for a maximal Poincaré inequality is also considered. In this case the left-hand side in the Poincaré inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincaré inequalities is used to characterize the self-improvement of two-measure Poincaré inequalities. Examples are constructed to illustrate the role of the assumptions. Harmonic analysis and PDE techniques are used extensively in the arguments.

Original languageEnglish
Pages (from-to)1763–1810
JournalJOURNAL OF GEOMETRIC ANALYSIS
Volume29
Issue number2
Early online date2018
DOIs
Publication statusPublished - Apr 2019
MoE publication typeA1 Journal article-refereed

Keywords

  • Geodesic two-measure space
  • Poincaré inequality
  • Self-improvement

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